# Space Travel - Lorentz Transformation

Space ship Alpha travels at t = 0 and v = 4/5 c to the star Sirius which is 8.6 light years away. One year later space ship Delta starts at v = 9/10 c to the same star.

Question 1:
When does Delta overtake Alpha, as measured from Alpha's, Delta's and Earth's perspective?

Question 2:
At which distance to Earth (measured from Earth's system) does this happen?

Regarding 1:
For Earth's perspective I use $$\delta t = \frac{1}{\sqrt{1-\beta^2}} * \delta t_0$$. Using that I get to 17.92 years for Alpha and 21.92 years (plus an additional year because the ship left a year later) for Delta. This would mean that from Earth's perspective Delta never passes Alpha.
For Alpha's perspective I use length contraction for the 8.6 light years and then use v = s/t to get the time for 4/5 c and the shortened distance. For Delta's perspective I use the same approach.

Would this be correct so far?

Now my question is regarding Alpha's speed as seen from Delta's perspective and vice versa. Would I just use the speed of the ship relative to the other for this and then also include length contraction? I'm not quite sure how to do this otherwise.

Regarding 2:
From Earth's perspective Delta never overtakes Alpha. In the other systems I guess Delta would pass Alpha. But since there's length contraction involved how would I approach this? Would I just calculate where that point is relative to the entire distance (meaning in %) and then apply that to the distance as seen by Earth?